↑

Properties

Label 1-2432-2432.1299-r1-0-0
Degree $1$
Conductor $2432$
Sign $0.980 + 0.194i$
Analytic cond. $261.354$
Root an. cond. $261.354$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.997 − 0.0654i)3-s + (0.946 + 0.321i)5-s + (−0.382 − 0.923i)7-s + (0.991 − 0.130i)9-s + (−0.831 − 0.555i)11-s + (−0.946 + 0.321i)13-s + (0.965 + 0.258i)15-s + (−0.965 + 0.258i)17-s + (−0.442 − 0.896i)21-s + (0.130 + 0.991i)23-s + (0.793 + 0.608i)25-s + (0.980 − 0.195i)27-s + (0.896 + 0.442i)29-s + i·31-s + (−0.866 − 0.5i)33-s + ⋯
L(s)  = 1  + (0.997 − 0.0654i)3-s + (0.946 + 0.321i)5-s + (−0.382 − 0.923i)7-s + (0.991 − 0.130i)9-s + (−0.831 − 0.555i)11-s + (−0.946 + 0.321i)13-s + (0.965 + 0.258i)15-s + (−0.965 + 0.258i)17-s + (−0.442 − 0.896i)21-s + (0.130 + 0.991i)23-s + (0.793 + 0.608i)25-s + (0.980 − 0.195i)27-s + (0.896 + 0.442i)29-s + i·31-s + (−0.866 − 0.5i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $0.980 + 0.194i$
Analytic conductor: \(261.354\)
Root analytic conductor: \(261.354\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (1:\ ),\ 0.980 + 0.194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.589384929 + 0.3532676385i\)
\(L(\frac12)\) \(\approx\) \(3.589384929 + 0.3532676385i\)
\(L(1)\) \(\approx\) \(1.613117894 - 0.03409367173i\)
\(L(1)\) \(\approx\) \(1.613117894 - 0.03409367173i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.997 - 0.0654i)T \)
5 \( 1 + (0.946 + 0.321i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.831 - 0.555i)T \)
13 \( 1 + (-0.946 + 0.321i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (0.896 + 0.442i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (0.793 - 0.608i)T \)
43 \( 1 + (-0.442 - 0.896i)T \)
47 \( 1 + (0.258 - 0.965i)T \)
53 \( 1 + (-0.0654 + 0.997i)T \)
59 \( 1 + (0.946 + 0.321i)T \)
61 \( 1 + (-0.442 + 0.896i)T \)
67 \( 1 + (0.442 - 0.896i)T \)
71 \( 1 + (-0.991 - 0.130i)T \)
73 \( 1 + (0.608 + 0.793i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + (-0.980 - 0.195i)T \)
89 \( 1 + (-0.793 - 0.608i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.44581043243724437367719883910, −18.49145999672529790668702355836, −18.07648144343181135680617760679, −17.30698990707702553768854177851, −16.26271518510126847966581241073, −15.692947192118978042440907578756, −14.888231032318731333691926636782, −14.45761245360191623158833880329, −13.42083098749551369233588410541, −12.87356294491619490199294986028, −12.52423171505726591322290744294, −11.33347441846117682436285909974, −10.14058834798647387125061832297, −9.8048371609071174691488873381, −9.13945341258360525319266441674, −8.387856500313961028303913868541, −7.690326553135896759940051900938, −6.67153354443853023790022518029, −5.97594441286245222665161328382, −4.841165752953832862670753146568, −4.53161696573344272947816679058, −2.95449140133915441159012758731, −2.47569398901678391725117506146, −2.015494039387820360457498939298, −0.60058304817335904462049114358, 0.76772316723047924524063937616, 1.823126789000774314842738105981, 2.60018302661796686113333603520, 3.24171897351291577359377594150, 4.20805335739415668699921786539, 5.064579061571104688581631983426, 6.07514612519119449904508235847, 7.080213741548755329052846900727, 7.30606060026992179847728876215, 8.4714468829186800972349016181, 9.112860660072839933448953602525, 9.96114774853537458149410711462, 10.36647543677941962483243999059, 11.15081059525041250976993947756, 12.4766350329664063196687550173, 13.09232365566554308912677309381, 13.80987826254655707408323395366, 14.02490984136478097955730487477, 14.98339266936343518526221972780, 15.69862281675539978547516923520, 16.49888393600621192862555955262, 17.31205743761275755593930322690, 17.978078408529099798558660663704, 18.685418605365676441699213036698, 19.60918354676403861906508296448

Graph of the $Z$-function along the critical line